$B-L$ violating nucleon decays as a probe of leptoquarks and implications for baryogenesis

We study the effective $B-L$ violating couplings for scalar and vector leptoquarks which can naturally induce dimension seven $B-L$ violating operators leading to very interesting nucleon decay modes such as $n \rightarrow e^- \pi^+, e^-K^+,\mu^- \pi^+, \mu^-K^+$ and $p \rightarrow \nu \pi^+$. This opens a new window to probe the nature and couplings of the scalar and vector leptoquarks in an ultraviolet model independent way which can provide an orthogonal probe for scalar and vector leptoquark solutions to the recent anomalous $B$-decay data. Furthermore, we also discuss how these $B-L$ violating interactions can also pave a new way to understand the origin of matter-antimatter asymmetry of the Universe.


I. INTRODUCTION
The renormalisable Standard Model (SM) Lagrangian interactions cannot violate baryon number, and consequently, baryon number violation can only arise through effective higher dimensional operators with dimension six or higher [1][2][3], making baryon number violation a very sensitive probe of new physics beyond the SM. The leading dimension six operators can naturally be generated in many grand unified theory (GUT) embedding of the SM and they conserve baryon number minus lepton number (B − L) symmetry leading to proton decay modes such as p → e + π 0 , p → µ + π 0 and p → νK + , which have been the primary focus of the recent experimental searches. On the other hand, d = 7 operators obeying the selection rule ∆(B − L) = −2 [4,5] leading to interesting decay modes such as n → e − π + , e − K + , µ − π + , µ − K + and p → νπ + have received considerably less attention. These B − L violating decay modes have been discussed in the context of G Pati-Salam , and SO (10) GUT theories in Ref. [6] and in the context of SO(10) GUT theories in Refs. [7,8]. In these studies, driven by the GUT theory motivations, the mass scales of the mediating particles have been taken to be very heavy ∼ 10 11 − 10 13 GeV. In this work we point out that these B − L violating nucleon decay modes provide a novel way to probe the nature and couplings of the scalar and vector leptoquarks in a model independent way 1 .
This not only makes these B − L violating nucleon decay modes extremely interesting for current and future experimental searches, but also gives way to a new type of mechanism to understand matter-antimatter asymmetry of the Universe where simultaneous baryon and lepton number violation gives rise to a B − L violating asymmetry, which unlike B − L conserving asymmetry, does not get washed out by B + L violating sphaleron interactions near the electroweak scale [10]. Furthermore, this also motivates the experimental search for B − L violating nucleon decay modes as an orthogonal probe for TeV scale scalar and vector leptoquark solutions to the B−decay anomalies, which are persistent with new data from the B−factories [11][12][13][14][15][16][17][18][19][20][21][22][23], and are currently one of the strongest hints of new physics beyond the standard model. To this end, the recent measurements of the ratio of branching fractions of B → K(K * ) decays into di-muons over di-electron modes R K ( * ) and the ratio of branching fractions of B → D ( * ) −ν decays into tau over other lepton modes R D ( * ) are of particular interest because in these ratios the hadronic uncertainties cancel and consequently, these observables are sensitive to lepton flavour universality (LFU) violating new physics.
These anomalies have resulted in a plethora of very interesting studies, among which two very popular scenarios which can address the above anomalies are the extensions of the SM with scalar leptoquarks and vector leptoquarks . Vector leptoquark couplings have the potential to simultaneously explain both R K ( * ) and R D ( * ) [81], however a minimal G Pati-Salam GUT embedding of such vector leptoquark is inconsistent with very strong bounds from various flavour violating processes. Consequently, non-minimal extensions of such embedding is required to avoid such bounds [80][81][82][83][84][85][86][87][88][89][90][91][92][93][94]. On the other hand, single scalar leptoquark cannot simultaneously address R K ( * ) and R D ( * ) data without violating constraints from various flavour violating processes and consequently attempts have been made to use more than one leptoquark and to embed them in minimal GUT frameworks to such end [63,95]. In this work we will primarily focus on the scalar leptoquark states tabulated in Table. I and the vector leptoquarks U 1 : (3, 1, 2/3) andṼ 2 : (3, 2, −1/6).
The plan for rest of this paper is as follows. In section II, we discuss the effective leptoquark couplings and B − L violating operators with d = 7 and provide the relevant diagrams and estimation for the lifetime for nucleon decay modes. In section III, we discuss possible UV completions for these interactions and the possible origins of the B −L violating couplings in the context of these UV completions. In section IV, we discuss the possible connection between the B-decay anomalies and the B − L violating nucleon decay rates. In section V, we explore the implications of these B − L violating interactions for baryogenesis and the possible correlation between the final baryon asymmetry and the nucleon decay lifetime. Finally, in section V we summarise and make our concluding remarks.
where i, j, k, l are SU (2) L indices, and all the fermion fields are written in terms of left handed spinors. We will first show that some of these operators can arise naturally in the presence of effective trilinear couplings involving two scalar (vector) leptoquarks and the SM Higgs doublet. Interestingly, at the effective level the coupling coefficient of these trilinear interactions have mass dimension unity and is proportional to the B − L breaking scale, which can in principle be decoupled from the gauge coupling unification scale in a given UV complete model. We will first treat the problem at an effective level where we will only fix the effective mass scales corresponding to the coupling coefficient of this trilinear interactions and the mass scales of the leptoquark states to show that these operators can give rise to nucleon decay rates which can be probed at the current and future experimental nucleon decay search facilities. The relevant effective trilinear scalar interactions invariant where i, j, k are SU (2) L indices. To see the B − L assignments of the scalar leptoquark states and the above couplings let us write down the Yukawa interactions for the relevant scalar leptoquarks at an effective level- where c corresponds to the SU 2 Note that one can also have d = 12 nucleon decays in the presence of trilinear couplings involving three scalar leptoquarks [96] and trilinear couplings involving a triplet Higgs scalar, a singlet and a triplet leptoquark [97]. of Eq. (1) through the effective coupling where the vector leptoquarks couple to the SM fields through the interactions The couplings in Eq.
Consequently, Eq. (7) together with the Yukawa interaction given in Eq. (8)) lead to ∆(B − L) = 2 nucleon decay. In Fig. 1. we show some representative diagrams inducing d = 7 B − L violating nucleon decay operators in the presence of the scalar and vector leptoquark couplings discussed above. For illustration purposes we will consider the simple case of partial lifetime for the decay mode n → e − π + . The lifetime for this decay mode induced by the trilinear coupling involving scalar leptoquarks P 1,2 and H, can be estimated by [98] where λ correspond to the dimension-ful B − L violating trilinear coupling of P 1,2 and H; Y P 1 and Y P 2 are the Yukawa couplings for the scalar leptoquarks P 1 and P 2 . v ≡ H 0 = 174 GeV, β H 0.012 GeV 3 is the nucleon decay matrix element and where the D and F correspond to contributions from chiral Lagrangian factors. The decay rate induced by the effective coupling involving vector leptoquarks U 1 ,Ṽ 2 and H given in Eq. (10) is given by Here x U 1 and xṼ 2 are the couplings of U 1 andṼ 2 with SM fields, [99,100].

III. UV COMPLETION AND ORIGIN OF B − L VIOLATING COUPLINGS
At an effective level the computation of the decay rates given in Eqs. (9) and (10)  The effective B − L violating couplings can naturally arise in GUT theories like SO(10) and In such theories the effective B − L violating couplings given in Eqs. (2) and (7) can arise when the local B − L symmetry is broken by giving a vacuum expectation value to the SM singlet field ∆ c carrying B − L = −2. ∆ c is present in the 126 H multiplet of SO (10) and corresponds to (1,3,10) under G Pati-Salam [7,8]. Note that in SO(10) GUT ∆ c can also generate large Majorana masses for the right handed neutrinos through the couplings of the form ν c ν c ∆ c .
(4), (5) and (6) where parenthesised superscripts have been used to distinguish more than one fields with the same quantum numbers appearing in some decompositions. In principle, after the GUT Interestingly, the relevant effective Wilson coefficients can get significant contributions from where λ t = V tb V * ts . The most relevant operators are at the 2 σ (1 σ) level 3 .
The possibility of accommodating such a scenario for S 3 leptoquark in the context of SU (5) GUT completion has been discussed in Ref. [103,104] and it has been explicitly pointed out that the term 10 i 10 j 45 H gives rise to the di-quark coupling for S 3 and consequently, such a coupling must be absent to avoid constraints from d = 6 proton decay if S 3 leptoquark mass is around TeV scale. Here we will be interested in the SO(10) breaking to the SM gauge group through the intermediate subgroup From the SO(10) GUT completion perspective, from Eq. (14) we note that the coupling 16 i 16 j 120 H would lead to d = 6 rapid proton decay due to the presence of both di-quark and leptoquark couplings of S 3 leptoquark and therefore is inconsistent with a TeV scale S 3 explaining R K ( * ) anomalies. More interestingly, we note that in the coupling 16 i 16 j 126 H the possibility of such a rapid proton decay due to d = 6 operator induced by S 3 is absent and furthermore, we also notice that R 2 (which can possibly explain resolve R D ( * ) anomalies for m R 2 ∼ 1 TeV) can be embedded in such a coupling without inducing any rapid nucleon decay modes. Therefore, we will focus on this scenario where As a benchmark point taking MR 2 ∼ 1 TeV, M S 1 ∼ 10 16 GeV, y S 1 ,R 2 ∼ 10 −3 we find from Eq. (9) that the lifetime τ ≈ 3×10 33 yrs for λ 2 = 10 11 GeV, which is in the observable range For vector leptoquarks we note that a TeV scale U 1 can address the anomalies in b → s and b → c ν data simultaneously. However, the strong bound from charged lepton flavour violating decay processes can only be satisfied if U 1 couples to the SM fields trough a mixing with vector-like counterparts of the SM fields and moreover the first generation mixing is further suppressed to account for the bounds from K L → µe and K → πµe data (e.g. by some additional flavour symmetry) [82][83][84]94]. To account for such a suppression we parametrise x U 1 ,Ṽ 2 = βg U where β ∼ 10 −3 . Taking M U 1 ∼ 1 TeV and MṼ 2 ∼ 10 16 GeV (as it can also mediate d = 6 nucleon decay) we obtain from Eq. (10) the lifetime τ ≈ 1.   Fig. 2 we show the one loop diagrams which can interfere with the tree level decay modes of S 1,3 andṼ 2 to provide the CP violation. The computation of the final asymmetry is UV model dependent and must also take into account the potential gauge and Yukawa washout processes [115]. In the absence of more than one generation of S 1 only the top two diagrams in Fig.   2 contributes and the contribution coming from these diagrams are proportional to the Majorana masses for ν c . Assuming the Majorana mass matrix of the ν c fields to be diagonal and real, the asymmetry contribution coming from these two diagrams is given by Here we define the Yukawa coupling matrix corresponding to the coupling ν c LH, as y ν c LH and M ν c is the real diagonalised mass matrix of ν c fields. f r corresponds to the branching where Θ stands for the step function, accounting for additional ways of cutting the diagram for the case M ν c < MR 2 . In addition to the above contributions, another additional contribution can arise in any realistic UV complete model where there are more than one S 1 fields. Denoting the heavier S 1 as S 1 the contribution coming from the bottom-left diagram of Fig. 2 is given by 6 where the loop function G( For the case where S 1 and S 1 are degenerate in mass, a resonant enhancement is possible. For a prescription of such a case see for example Ref. [116]. 5 The interplay between the (B − L)-conserving decays and the (B − L)-preserving decays of leptoquarks are crucial for inducing generating the asymmetry as first pointed out in Refs. [7,8]. 6 We assume M S1 − M S 1 Γ S1 , so that there is no resonant enhancement for the decay.
The branching ratio f r ≡ Γ(S † 1 →R 2 H * )/(Γ(S † 1 →R 2 H * ) + Γ(S † 1 → f f )) in the asymmetry can be estimated using the partial decay widths The baryon asymmetry to entropy ratio Y B is given by where g * is the total number of relativistic degrees of freedom and d in Eq. (25) is the dilution factor taking into account the partial wash out processes, which can be obtained exactly by solving the Boltzmann equation. The dilution factor can be approximately estimated by where with H denoting the Hubble expansion rate given by In Fig. 3, we show the dependence of the final baryon asymmetry on the couplings for y S 1 ∼ yR 2 , for different values of M ν c and y ν c LH , while choosing the benchmark values for rest of the parameters: MR 2 = 1 TeV, M S 1 = 10 16 GeV, λ = 10 11 GeV and y f S 1 = 10 −3 . A direct correlation between the final baryon asymmetry and the lifetime for nucleon decay mode can also be obtained under the simplifying assumption that the leptoquark dominantly couples to first generation SM couplings 7 . In Fig. 4, we show the correlation between the final baryon asymmetry and the partial lifetime for the decay mode n → e − π + for different values of M ν c and y ν c LH with the same benchmark values for the rest of the parameters as in Fig. 3. It is important to note that various partial wash out processes can occur